(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2005)
Let v_1,…,v_r be a β-sequence (Definition 1.2) in an n-vertex graph G and v_r+1,…,v_n be the other vertices of G. In this paper we prove that if v_1,…,v_r is balansed, that is \[ 1r(d(v_1)+… +d(v_r)=1n(d(v_1)+… +d(v_n),\] and if the number of edges of G is big enough, then G is regular.